Centre for Market and Public Organisation Bristol Institute of Public Affairs
University of Bristol 2 Priory Road Bristol BS8 1TX http://www.bristol.ac.uk/cmpo/ Tel: (0117) 33 10799 Fax: (0117) 33 10705 E-mail: cmpo-office@bristol.ac.uk
The Centre for Market and Public Organisation (CMPO) is a leading research centre, combining expertise in economics, geography and law. Our objective is to study the intersection between the public and private sectors of the economy, and in particular to understand the right way to organise and deliver public services. The Centre aims to develop research, contribute to the public debate and inform policy-making.
CMPO, now an ESRC Research Centre was established in 1998 with two large grants from The Leverhulme Trust. In 2004 we were awarded ESRC Research Centre status, and CMPO now combines core funding from both the ESRC and the Trust.
ISSN 1473-625X
An Economic Theory of Glass Ceiling*
Paul A.Grout, In-Uck Park and Silvia Sonderegger
October 2009
An Economic Theory of Glass Ceiling*
Paul A. Grout
,In-Uck Park
andSilvia Sonderegger
CMPO, University of BristolOctober 2009
*
(Updated from working paper 07/183)Abstract
In the ‘glass ceiling’ debate there appear to be two strongly held and opposing interpretations of the evidence, one suggesting it is really the result of gender differences and the other that there is discrimination by gender. This paper provides an economic theory of the glass ceiling and one of the main insights of our analysis is that in some real sense these two interpretations are not in conflict with each other. The glass ceiling emerges as an equilibrium phenomenon when firms compete à la Bertrand even though employers know that offering women the same contract as men would be sufficient to erase all differences among promoted workers. The model also provides new insights into anti-discrimination policy measures.
Keywords:glass ceilings, promotions, career options JEL Classification: J16, D82
www.bristol.ac.uk/cmpo/publications/papers/2009/wp227.pdf
Electronic version:
Acknowledgements
We thank Dan Bernhardt, Martin Cripps, Spyros Dendrinos, Bart Lipman, Martin Hellwig, Ian Jewitt, Glenn Loury and Andy McLennan for useful conversations and comments. We also thank seminar participants at Boston, Brown, Korea and Kyoto Universities and the Universities of Bonn, Bristol, SUNY at Albany, and Queensland, and the ESEM 2008 in Milan. Any remaining errors are our own. The authors thank the Leverhulme Trust for funding this research project.
Address for Correspondence
CMPO, Bristol Institute of Public Affairs University of Bristol 2 Priory Road Bristol BS8 1TX p.a.grout@bristol.ac.uk i.park@bristol.ac.uk s.sonderegger@bristol.ac.uk www.bristol.ac.uk/cmpo/
Paul A. Grout, In-Uck Park, Silvia Sonderegger
University of Bristol, U.K. October 28, 2009
Abstract. In the ‘glass ceiling’ debate there appear to be two
strongly held and opposing interpretations of the evidence, one sug-gesting it is really the result of gender differences and the other that there is discrimination by gender. This paper provides an economic theory of the glass ceiling and one of the main insights of our anal-ysis is that in some real sense these two interpretations are not in conflict with each other. The glass ceiling emerges as an equilibrium phenomenon when firms compete `a la Bertrand even though employers know that offering women the same contract as men would be suffi-cient to erase all differences among promoted workers. The model also provides new insights into anti-discrimination policy measures. (JEL Codes: J16, D82)
Keywords: Glass ceiling, promotions, career options.
1. Introduction
The glass ceiling is one of the most controversial and emotive aspects of em-ployment in organizations. The term appears to have originated only in the mid 1980s but became so rapidly sealed in the lexicon that by 1991 the US had created a Federal Glass Ceiling Commission with the Secretary of Labor as its chair. When setting up the Glass Ceiling Commission in 1991 the US Department of Labor defined the concept as “those artificial barriers based on attitudinal or organizational bias that prevent qualified individuals from advancing upward in their organizations into management-level positions”. It added that these barriers reflect “discrimination ... a deep line of demarcation between those who prosper and those who are left behind.” One only has to look at the casual empirical evidence to see why the issue remains topical and heated.
Women form a disproportionately small group in senior management posi-tions. For example, Figure 1 provides the proportion of females in employment
∗We thank Dan Bernhardt, Martin Cripps, Spyros Dendrinos, Bart Lipman, Martin Hell-wig, Ian Jewitt, Glenn Loury and Andy McLennan for useful conversations and comments. We also thank seminar participants at Boston, Brown, Korea and Kyoto Universities and the Universities of Bonn, Bristol, SUNY at Albany, and Queensland, and the ESEM 2008 in Milan. Any remaining errors are our own. The authors thank the Leverhulme Trust for funding this research project. (Emails: p.a.grout@, i.park@, s.sonderegger@bristol.ac.uk)
amongst US professions and the proportion of female within those employ-ees working as officials and managers (US Equal Employment Opportunity Commission). Women constitute just over half of all professions but little more than a third of all officials and managers. In the US Fortune 500 women account for only 15.6% of all corporate officer positions of any type. Further-more, not only do women form a minority of employees at senior levels, they also receive lower remuneration than men. This disparity is reflected through-out senior management. Figure 2 shows the relative salary of educated women (according to highest education attainment) to equivalent educated men in the US from 1990 to date. The ratio for both women with bachelors de-gree and those with an advanced qualification (i.e., higher than bachelors) are relatively constant and very similar, with mean values below 0.6 across the period.
In the current debate, and to a lesser extent in the academic literature, there appear to be two strongly held and opposing interpretations of all this evidence. One interpretation is that this is the result of real gender differences. The other interpretation is that what is observed is only consistent with dis-crimination by gender. This paper provides an economic theory of the glass ceiling. One of the main insights emerging from our analysis is that in some real sense these two views are not in conflict with each other. A critical as-sumption of the model is that there is more diversity in women with regard to job commitment than men1 (we discuss this later in this section) who, as a result, have less potential for private information. In this sense, consistent with the first interpretation, gender differences are at the heart of the story. However, in our model, employers know that if female workers were offered the same promotion contracts as men, all promoted women would display the same job commitment as men, and so this private information would become irrelevant. Thus, if any difference persists between male and female promotion contracts in equilibrium nonetheless, it cannot be justified by invoking differ-ences in the two genders’ propensity to leave their high-rank jobs. In spite of this, we show that in equilibrium firms choose to offer promotion contracts that differ between the two genders, even when they compete `a la Bertrand for employees. So, in a real sense the second interpretation is also correct. The model allows us to understand the ways in which the discrimination in-terpretation is correct and gives insight into the design of anti-discrimination policies.
There are papers that find pure gender discrimination. Goldin and Rouse
1This is backed for instance by recent evidence by Bertrand, Goldin and Katz (2009),
who have tracked the careers of MBAs who graduated between 1990 and 2006 from a top U.S. business school. They show that the presence of children is the main contributor to greater career discontinuity and shorter work hours for female MBAs relative to their male counterparts.
(2000) look at the effect of symphony orchestras choosing to use “blind” audi-tions that conceal the musician. They find blind audiaudi-tions increase the prob-ability that women are hired. Similarly, Neumark et al (1996), using matched groups of men and women, find evidence of gender discrimination in hiring in restaurants. On the other hand, there is also a considerable literature on gender differences and associated economic disparities both within economics and from outside the discipline. For example, Babcock and Laschever (2003) argue that women are poor negotiators and generally dislike the process of negotiating. Browne (1995, 1998) suggests that men are more interested in striving for status in hierarchies and “engage in risk taking behavior that is of-ten necessary to reach the top of hierarchies.” Kanazawa (2005) uses General Social Survey data to show that men rank financial reward and power posi-tions much higher in their preferences for employment, concluding that since men covet and strive for such positions they are the ones who are more likely to succeed in achieving them, whereas “women have better things to do.” Some recent economic experiments are related to this literature. Gneezy et al (2003) find that men perform significantly better than women in a more com-petitive, tournament environment, while there is no gap in a non-comcom-petitive, piece rate environment. When subjects can choose between a piece rate and a tournament environment, Niederle and Vesterlund (2007) find a significant gender gap in choice, with 35% of women and 73% of men selecting the tour-nament. Fryer et al (2008) report that the introduction of financial incentives (as opposed to increased competition) exacerbates the gender gap by enhanc-ing the performance of men significantly, while it increases the stress levels of women more than men, in their experiment.
Economists have studied differential treatments in labor market from the perspectives of discrimination theories, pioneered by Becker (1959), Arrow (1973) and Phelps (1977), and further developed by other authors, e.g., Coate and Loury (1993). The current paper, however, explains the glass ceiling as a competitive equilibrium outcome of agents dealing with informational problems. Hence, our approach is more closely related to Lazear and Rosen (1990) who study gender differential treatment by non-discriminating employ-ers. Similar to them, we study an environment where women have non-market opportunities that are on average better than those for men. There are, how-ever, important differences. In Lazear and Rosen, the fact that women face worse promotion prospects than men is somewhat “justified” since promoted women do have on average a higher propensity to leave or take a career break than men. By contrast, in our approach firms know that they can offer con-tracts that would remove such gender differential conditional on promotion. Specifically, in our equilibrium the promotion offer made to men would domi-nate the non-market option for all promoted women, rendering them identical to promoted men because they would never leave. This implies that there is
no a priori reason why women should be made promotion offers differently from men. Nonetheless, we show that women are consistently offered inferior promotion deals than men, earning a lower wage in high-rank jobs. This dif-ferential treatment is “unjustified” since promoted women are no worse than men in productivity. Relative to Lazear and Rosen, our model shows that the glass ceiling may emerge as an equilibrium phenomenon of the labor market competition, even though it is not grounded in productivity differences.
The formal model we consider is loosely as follows. There are two tech-nologically identical firms competing for employees.2 Employees are hired to entry-level positions and each firm has need for one senior management slot for every two entry-level positions. The amount of effort (human capital invest-ment) that employees put in at entry-level positions affects their productivity in senior-level posts if they get promoted, but has no impact on their output in the second period otherwise.
The firms compete `a la Bertrand in the labor market at an initial hiring stage by offering contracts that specify their terms of promotion. We analyze what happens when contracts can be made gender specific with a view to understanding the impact on the equilibrium when various possible legal anti-discrimination restrictions are placed on the market. Employees approach the firm that offers the best contract for them, and then take up the other offer if they are not chosen by that firm.
Firms select their senior level managers from the intake at the lower level in the previous period based on the effort levels they exerted and the contracts offered. Since we assume that these effort levels are not observable by the rival firm, promotions take place internally within the firm in equilibrium. Ex-post negotiations are allowed between a firm and workers promoted in the rival firm, but it does not happen in equilibrium for it would necessitate reneging on a contract with some other worker. Note that even if the effort levels were observable by rival firms, there are good reasons why ex-post negotiations may not matter. For instance, it would be so if we introduced a small firm-specific element in human capital investment into the model.
We assume that women differ in terms of the non-market options available to them. Whether these options are worthwhile to pursue when they arise, however, is endogenously determined. The equilibrium has two main features. The first is that promoted women end up being paid less than men, even though they are equally productive. This is a striking results since, as dis-cussed above, this differential treatment cannot be justified by invoking gender differences in productivity. Employers know that offering the male promotion contract to female workers would be sufficient to erase all gender differences
2We think of the two firm market being sustained by entry costs that limit the number
among promoted workers. Hence, in our model, the differences in the promo-tion offers made to male and females do not stem from productivity differences, but from other forces.
The second feature is that the differential treatment suffered by women in their promotion deals persists even when firms compete Bertrand-style for employees. Moreover, we show that even though firms are ex-ante identical and follow the same strategy, the equilibrim is asymmetric ex-post, in the sense that one firm is more female friendly than the other. Hence, in equilib-rium one firm will employ more women3 and extract a smaller surplus from them than the other firm.
What is the basic intuition why, even in the presence of Bertrand compe-tition, promoted women do not receive full reward for their effort? It starts from the observation that the option available for the workers who forgo pro-motion is worse for women than men due to the competitive labor market, because the compensation fully reflects the gender differences in commitment for those workers. This means that women with low non-market options are willing to accept worse promotion deals than men, since their market alter-natives are worse than those of men. Hence, a firm can, at least initially, promote women more “cheaply” than men. But, promoting more women is gradually more costly, since it requires inducing women with better and bet-ter non-market albet-ternatives to apply for promotion, and therefore it requires the promotion deals offered to women to become gradually more attractive. Bertrand competition raises firms’ bids for women workers and reduces the ex-tra surplus that promoted women generate. However, this competition stops short of erasing the extra surplus completely, because at some point it pays for a firm to abandon the bidding war and instead, extract the maximum surplus from the residual women workforce.4 This also explains why one firm is more female friendly than the other in equilibrium.
Thus the model provides an explanation for the glass ceiling phenomenon that, although not grounded in real productivity differences between men and women in high-rank posts, is the outcome of a competitive process. We are able to show that this equilibrium does not depend on being able to offer gender specific contracts. However, if promotion rules (such as minimum promotion ratios for women) are imposed on the organization, then the
equi-3It is worth noting that separation of black and white workers is taking place between
firms in Lang, Manove and Dickens (2005). However, their model is one of monopolistic competition in labor market without human capital investment and promotion decisions and thus, their focus and analysis are different from ours.
4Astute readers will have notice that this is not a full description of the equilibrium in
a simultaneous move game because if one firm abandons the bidding war, the other firm’s bid would not be optimal. For this reason, firms use a mixed strategy in equilibrium. In addition, see the discussion following Theorem 2 in Section 4.
librium can change to improve the position of those women who have a low propensity for career breaks. A central feature of our approach that separates our conclusions from other discrimination models is that these restrictions cannot be temporary incursions into the labor market to shift the equilibrium to a more favorable one in a multi-equilibrium environment.5 In our model, the interventions would have to be “permanent.”
However, we also show that there are significant differences between pro-motion rules, in particular, between rules that set lower bounds on the fraction of women to be promoted and rules on the fraction of senior posts that must be filled by women. In the former it is necessary to choose exactly the right bound (which is a measure zero event) to eliminate the glass ceiling problem, but in the latter setting any bound in an interval will do.
Before moving to the main body of the paper it is useful to highlight the potential applicability of the model. Although we have focused on gender, we see the model as being relevant for other observationally distinct groups as well. An immediate broad analogy is the treatment of immigrant employees (particularly where the culture of the country they have left is very different from the host country), since these foreign employees may wish to return to their home country at some point.6
The next section describes how we model the environment as a game. Section 3 analyzes the game when there is a monopoly employer, which is extended in Section 4 to characterize the unique equilibrium outcome under Bertrand competition between employers. Section 5 addresses policy implica-tions. Section 6 contains some concluding remarks.
2. Model
We consider a labor market with two populations of workers, male and female, of the same measure 2. There is an initial period of their employment, period 1, during which they decide on human capital investment. Depending on their investment decision, they may get one of two kinds of jobs in period 2, referred to as upper-tier and low-tier jobs as explained below. In the middle of period 2 all workers may encounter a non-market option with the same probability p, at which point separation may occur as detailed below. We assume that female
5Although many models have the feature that affirmative action only needs to be
tem-porary, this is not always the case (see Coate and Loury (1993)).
6An alternative anecdotal example (encountered by one of the authors) concerns a full
time employee who enjoyed writing novels in his spare time. He claimed that he always felt disadvantaged relative to his colleagues since it was always implied that he was not as committed to the career as others because “surely he would really prefer to be a full time author” and was only waiting for the opportunity. He claimed this question arose to differing degrees of directness at every appraisal he ever had, and that he thus felt obliged to display greater commitment to the cause than other employees.
workers are heterogenous in the value of their non-market options. Precisely, each female worker has a private type θ drawn from a commonly known cdf
D(θ) on R+, where θ is the value of her outside, non-market option she may
encounter while on the job. Men are much less varied in this dimension. In particular, for simplicity, we assume that all male workers are homogeneous in the sense that the value of their non-market option is zero (i.e., they do not encounter such options). For expositional ease, we assume that p = 1 and that θ is uniformly distributed over an interval [0, 1
α], i.e., D(θ) = αθ, where
α ∈ (0, 1/4) to ensure that the outside option is significant for a sufficiently
large fraction of female workers. Our results extend to large enough p < 1 and to a wide class of single-peaked distributions D, but exposition becomes more complex.
There are two firms, A and B, that are ex ante identical: each firm has a measure 2 of entry-level positions to fill in period 1, and a measure 1 of managerial positions, called upper-tier jobs, to fill in period 2. Note that it is not possible for both firms to hire only female (or only male) workers and, in particular, female workers are relatively scarce in this sense. This aspect is important for our results. However, the assumption that the total measure of the labor force is equal to total vacancies is not essential, and nor is that the ratio of upper-tier to entry-level posts is 1/2, and therefore, they can be relaxed but at a cost of expositional complication.
All workers (male and female) are assumed to have the same productivity in entry-level posts, which we normalize to 0. However, as indicated they can make a human capital investment/effort, e ∈ R+ in period 1, that would increase their productivity in the next period if they get promoted to an upper-tier post. The worker’s cost of making effort is quadratic7, c(e) = 1
2e2, and is incurred in period 1. The exerted effort level is observable only by the firm that he/she works for. Female workers learn their private types, θ, early in period 1, in particular, before their effort decisions.
In period 2, workers may get promoted to upper-tier posts. A promoted worker generates a flow revenue of y(e) = 1+e for the firm during period 2 (of length 1), where e is the effort exerted in period 1. However, at the midpoint of period 2, female workers encounter a shock that increases their outside option value from 0 to θ, in which case they choose whether (i) to remain in the job and get the contracted wage, or (ii) to leave the job (forfeiting the wage for the second half of period 2) and get the newly available outside option, which generates a flow (monetary equivalent) utility of θ for the remainder of period 2. We assume that if a worker leaves an upper-tier post then no revenue is generated by that post for the remainder of the period unless the worker is replaced by another worker who has held an upper-tier job in either firm.
All workers that do not get promoted can get a low-tier job in period 2, where they generate a constant flow revenue of 1 + κ after retraining which costs κ > 0 for the employer. The interpretation is that these jobs require “run of the mill” operations that can be carried out by anyone who has worked in an entry-level position. Since, unlike the upper-tier posts, all workers are equally productive while on low-tier jobs, competitive employers pay a flow wage that leaves them with zero expected profit: wm = 1 for men and wf < 1
for women. Here, wf < 1 reflects the market’s expected loss in revenue due
to the prospect of departure by female workers for the non-market option of
θ (if θ > wf) in the middle of period 2. For ease of exposition, we treat wf as
exogenous, although it can be determined endogenously in equilibrium, taking into account the types of women who get low-tier jobs, without affecting the main results. We present our analysis for wf ∈ (0.9, 1) to stress that our
results hold even when wf is arbitrarily close to wm (which is the case when
κ is arbitrarily small), but our results extend to a wider range of wf at the
cost of expositional complication.
Given that the competitive wages for low-tier jobs leave zero profit for the employer, it is inconsequential for our analysis whether such jobs are available in the firms that hired the workers initially or in a different section of the labor market. For simplicity, we present our analysis presuming the latter. The upper-tier posts are different in the sense that each firm has a comparative advantage of identifying more qualified workers among its own employees.
Therefore, prior to period 1, the two firms compete, `a la Bertrand, in at-tracting workers by offering more favorable (for the workers) contracts than their rivals. First, the two firms, i = A, B, publicly and simultaneously an-nounce their contracts si = (si
f, sim), consisting of one flow salary level sig ∈ R+ for each gender g ∈ {m, f } to be paid to workers promoted to upper-tier posts in period 2. We assume that the court enforces that no promoted worker gets paid less than the contracted salary rate si
g at any point in period 2 while on
the job. Thus, firm i’s “contract (offer) strategy” is represented by a proba-bility measure Fi on R2
+. Note that a contract does not specify a salary in period 1, which we assume is equal to the productivity, 0. This assumption is for ease of exposition and, as explained in Section 4, our main results extend to the case that a contract specifies period 1 salary as well.
After the contracts are offered, the workers are matched with the firms in an “allocation” stage. A precise modeling of this process, such as initial appli-cation and selection procedures and the second matching process of unfilled posts and residual labor supply, would involve nontrivial ad hoc assumptions. Hence, we take an alternative approach of directly postulating workforce al-location rules based on the fundamental principle that the firm offering a more favorable contract gets the first pick in hiring decisions. Here alloca-tions are represented by measures, µi
firm i ∈ {A, B}, such that µi
m+ µif = 2 for i = A, B, and µAg + µBg = 2 for
g = f, m. We denote µi = (µi m, µif).
If a firm, say i, has offered a contract si and hired µi = (µi
m, µif) of workers,
the continuation game has a nonempty set of equilibria (Lemma 6), denoted by E(si, µi). Let E(si) = ∪µiE(si, µi) and Π∗(si) be the maximum of the
firm’s equilibrium profit levels in E(si).
If firm i were to take the first pick in hiring, it would hire so as to maximize its profit in the continuation equilibrium. If there is a unique equilibrium in E(si) that generates Π∗(si) and it is the unique continuation equilibrium
following the firm’s offer of si and hiring µi of workers, then the firm will
indeed hire µi. In this case, the value of contract si is the female worker’s ex
ante utility in this equilibrium. Generally, the value of a contract si, denoted
by v(si), is the ex ante expected utility of a female worker in an equilibrium
of E(si) that generates Π∗(si) for the firm. We stress that v(si) is uniquely
defined in all contracts relevant for our main analysis, and the values of other contracts are inconsequential. Since all female workers prefer to be hired by the firm that has offered a contract with a higher value, we postulate that this firm has priority in hiring women as specified below. Note that v(si)
is defined endogenously to ensure that it correctly reflects the value female workers attach to a contract in equilibrium.8
The valuation of male contracts is different due to the homogeneity of male workers. Since they can guarantee a utility of 1 by exerting no effort and getting a low-tier job in period 2 (i.e., forgoing promotion), the value of any male contract is at least 1. On the other hand, even if some firm offers a very attractive male salary sm then as long as this firm hires more
than measure 1 of male workers, they would compete for promotion and, as a result, all promoted male workers end up exerting an effort level, say em, that
restores the equivalence of pursuing promotion and not, i.e., sm− e2m/2 = 1.
So, if either firm hires more than measure 1 of male workers, male workers know that they will end up obtaining a utility of 1, i.e., no higher than what they could obtain when they are hired by the other firm. This means that either firm is at least as well placed as the other firm in hiring additional male workers up to measure 1.
Based on these observations, we postulate the labor force allocation as9: (H1) The firm, say A, offering a contract with a strictly higher value first hires
8Alternatively, one could define the value exogenously albeit ad hoc, e.g., as the level of
female salary offered. Our qualitative results remain valid in this alternative definition.
9This allocation would ensue if the actual matching process is, for instance, as follows:
All workers apply to both firms (at no cost) and the firms have one chance of offering positions and the workers choose among the offered positions (randomly if equivalent offers), given that the firms have to fill all positions to operate.
as large a fraction of women as it wants, and as large a fraction of men as it wants up to measure 1. Then, firm B fills all its posts from the residual labor force. Finally, firm A hires any remaining workers. (H2) If the two firms offer contracts of the same value, they are allocated
measure 1 of each gender. Then, either firm may propose an alternative allocation, which is implemented if accepted by the other firm.
A strategy of each firm in an allocation stage, given the offered contracts
{sA, sB}, is a hiring decision as per the rule (H1) if v(sA) 6= v(sB), and a
decision as to which alternative allocation to propose and/or to accept as per (H2) if v(sA) = v(sB).
If v(si) > v(sj) we describe firm i as a leader (in hiring) and j as a follower.
Note that this is a slight abuse of terminology since, unlike the Stackelberg setting, the identities of firms are endogenous.
After contracts si are offered and workforce is allocated as µi for i = A, B,
a “promotion subgame” ensues, comprising periods 1 and 2. During these periods firms cannot fire workers, however workers may leave the firm at any time, forgoing any unpaid flow salary. At the beginning of period 1 all female workers learn their private types θ, and every worker in either firm exerts an effort level e ∈ R+, which is observable only by the employer. In period 2, each firm decides who to promote based on gender and exerted effort level, and pays them the salaries specified in the relevant contracts. All unpromoted workers leave the firm and get a low-tier job that pays a flow wage of wg.
After the firm has completed promotion decisions, we assume that any unpromoted worker may sue his/her employer. There is a case for a court to consider provided that he/she is potentially valuable for the firm to promote in the sense that 1 + e − si
g ≥ 0 where e is the exerted level of effort and g is
the gender of the worker. If the court verifies either
(Ci) that the firm promoted some worker who has exerted a strictly lower level of effort than the plaintiff, or
(Cii) that a non-zero measure of upper-tier posts are unoccupied,10
then the court finds in favor of the plaintiff and a hefty compensation payment must be made by the firm to the plaintiff. We assume a sufficiently high verifiability of effort levels in the court so that firms never leave any scope for a worker to successfully sue the firm.11
We note that verifiability of effort, which permits court protection pos-tulated above, is instrumental in obtaining uniqueness of equilibrium in our
10(Cii) is not necessary for our main result, but simplifies exposition considerably. 11Note that we are implicitly setting the cost of going to a court at zero, however, our
model, but not essential for the glass ceiling phenomenon. That is, our equi-librium (which exhibits a glass ceiling) continues to be an equiequi-librium, albeit no longer the unique one, even if effort is unverifiable (see Section 4).
At the midpoint of period 2, female workers encounter a non-market option of value θ, in which case they may leave the job to get θ. Here, we assume that when an employee is attracted to a non-market option that is more valuable than the current employment, departure is irreversible and renegotiation of salary is irrelevant at that point. A positive measure of such departures causes damage to the firm. For expositional ease, we capture such damage by simply postulating that the posts vacated by departures cannot be replaced. However, alternative modeling of what may happen to the vacated posts would not change our main results so long as the damage inflicted to the firm is nontrivial.12
In a promotion subgame of firm i with contract si and allocation µi, each
worker’s strategy consists of an effort level,13decision as to whether to sue the firm or not in the relevant contingencies, and for female workers, the decisions as to whether to leave employment for the non-market option of value θ; and firm i’s strategy specifies who to promote based on gender and exerted effort level (contingent on the profile of efforts exerted by all workers). Each worker maximizes the expected value of his/her income stream, including that from the non-market option and compensation from a lawsuit, net of any effort cost. Each firm maximizes its expected profit, i.e., total revenue net of total salary and lawsuit-compensation payments. We assume no discounting for simplicity.
A strategy profile in this promotion subgame, together with a belief pro-file on the type distribution of female workers contingent on the exerted effort level, constitutes a (perfect Bayesian) continuation equilibrium of this sub-game if the strategies are mutual best-responses and the belief profile satisfies Bayes rule whenever possible.
A strategy profile of the two firms in offering contracts, a strategy pro-file of the two firms in the allocation stage for each possible set of offered
12For example, the firms may try to fill vacated posts by recruiting workers in upper-tier
posts of the other firm by offering a higher salary. Then, a positive measure of departures in either firm would inevitably launch a recruiting war between the two firms, pushing the salaries of all upper-tier post workers up to their productivities (which the other firm can infer correctly in equilibrium), thereby depleting any positive profit of the firm for the remainder of period 2. Foreseeing this, the firms avoid hiring that would lead to departures and consequently, the equilibrium outcome is the same as in our model.
13Precisely, male workers’ effort choice is represented by a probability measure ξ
m on R+ and female workers’ effort choice by a probability measure ξf on R+× [0, 1/α] where
[0, 1/α] is the type space. Each ξgmay be interpreted either as the common mixed strategy adopted by all workers of gender g ∈ {f, m}, or the distribution of pure strategies adopted by measure µi
contracts {si}
i=A,B, and a profile of continuation equilibria for every possible
promotion subgame, constitute an equilibrium of the grand game if, given the profile of continuation equilibria, i) the strategy profile in the allocation stage contingent on {si}
i is an equilibrium in the continuation game, and ii) the
contract strategies of the two firms are mutual best-responses given the rest of strategies.
An equilibrium outcome of the game consists of a pair of (possibly stochas-tic) contracts {si}
i, allocations {µi}i contingent on {si}i, and the ensuing
effort profile of the workers and the promotion decisions, that arise in an equilibrium. We characterize the equilibrium outcome in the next two sec-tions.
3. Analysis of a Monopoly Firm
We start by analyzing the case where firm i is the only employer, because many of the core insights in this simpler environment carry over to the case of Bertrand competition. Then, firm i offers a contract, hires men and women as it wishes and a subsequent promotion subgame ensues (without presence of another firm). Consider a promotion subgame after firm i has offered a contract si and hired measure µi = (µi
m, µif) of workers where µim + µif = 2,
which we denote by (si, µi)-subgame. Since no promotions would take place
if upper-tier salaries are lower than the wage for low-tier jobs, it suffices to consider the cases that si
g ≥ wg, g = m, f , with at least one strict inequality.14
Consider a promoted worker in an arbitrary equilibrium of (si, µi)-subgame.
If this worker is male, it is clear that the firm would not pay him more than si m
because paying more would only incur higher expense without any benefit. If this worker is female and she does not leave in the middle of period 2, the firm realizes at that point that her outside option is no longer relevant and thus would pay no more than si
f for the remainder of period 2. Foreseeing this, she
would leave for her outside option if and only if θ > si
f, i.e., her decision to
leave depends only on the contracted salary level si
f. Understanding all this,
the firm would pay her no more than si
f throughout period 2. This
estab-lishes that any promoted worker would be paid exactly the contracted salary while on the job. It then follows, as the next lemma states, that all promoted workers of the same gender will have exerted the same level of effort and that no rationing of promotion takes place among workers who extended non-zero effort. The basic idea behind this is that rationing in promotion would be broken by some workers exerting a slightly higher effort level to break the tie and ensure promotion; and no worker would exert a higher effort than any other, indistinguishable worker for promotion in the absence of rationing in promotion.
Lemma 1. In any equilibrium of (si, µi)-subgame, all promoted workers
of gender g will have exerted the same effort level, say eg, and get paid exactly
the contracted salary, si
g, while on the job. In addition, if eg > 0, all workers
of gender g who exert eg get promoted, and all other workers exert e = 0.
Proof. It has been shown above that all promoted workers are paid exactly
the contracted salary while on the job. To reach a contradiction, suppose there is rationing in promoting workers who exerted a certain effort level, say e0 > 0.
Note that e0 is the lowest effort level that workers may get promoted with,
for otherwise (Ci) implies that the firm would lose the lawsuits filed by those who are rationed out after exerting e0. This also implies that the measure of
workers who exert an effort level strictly above e0, if exist, is strictly less than
1. Note that 1 + e0− si
g ≥ 0 if workers of gender g are promoted after exerting
e0, since otherwise the firm would not have promoted them. Thus, a worker
who deviates by exerting a slightly higher effort than e0(instead of e0), benefits
by warranting him/herself sure promotion owing to (Ci)-(Cii) because e0 > 0
implies si
g > wg. Since this would contradict e0 being an equilibrium effort
level, we conclude that any rationing, if it exists, would be among workers who exerted e = 0 and it would be possible only if they will get the same level of compensation promoted or not (for otherwise they would benefit by exerting small e > 0 for the same logic as above). Then, there is no more than one effort level for each gender, say eg, that leads to promotion, since
all workers prefer lower effort conditional on the same salary afterwards. Any worker who does not exert this effort level would not be promoted, so exerts
e = 0.
In an equilibrium of the (si, µi)-subgame, the effort level that workers of
gender g exert for promotion, which we denote by eg, determines their
deci-sions as to whether to pursue promotion or not, and the firm’s surplus from promoting them. For notational ease, we denote si = (s
m, sf) in this section.
Male workers would pursue promotion only if sm − (em)2/2 ≥ 1, with
certainty if the inequality is strict, because the payoff they can guarantee themselves from forgoing promotion is 1. Subject to this constraint, the firm’s surplus per male promotion, 1 + em− sm, is uniquely maximized at em = 1
and sm = 1.5, i.e., when the efficiency is achieved by equating the marginal
revenue and the marginal cost of effort. This proves
Lemma 2. The maximum possible equilibrium surplus of a firm per male
promotion is 0.5, which is possible if and only if em = 1 and sm = 1.5.
As will become clear below, these are the equilibrium terms of promotion for men.
private types θ. If a woman pursues promotion, she would get a utility of uh(θ|ef, sf) := max n sf, sf + θ 2 o − (ef)2/2 (1)
because she would leave the post for her outside option at midpoint of pe-riod 2 if θ > sf. If she forgoes promotion, she would exert no effort, get a
low-tier job for a wage of wf < 1, and would leave the employment for a
non-market option at midpoint if θ > wf, which warrants an expected utility of
u`(θ) := max n wf, wf + θ 2 o . (2)
Thus, a female worker of type θ would pursue promotion if uh(θ|ef, sf) > u`(θ)
and forgo promotion if the reverse inequality holds.
Whether women of types θ > sf would pursue promotion is critical for
the firm’s surplus because any such woman, by leaving at mid-career, would curtail any revenue that might be forthcoming otherwise. To understand when this happens and when not, observe that the graph of u`(θ) is flat at the level
of wf for θ ≤ wf, then increases with a slope of 1/2 for θ > wf. Similarly,
that of uh(θ|ef, sf) is flat at the level of sf − (ef)2/2 for θ ≤ sf and increases
with a slope of 1/2 for θ > sf.
First, if sf − (ef)2/2 < wf in equilibrium, the graph of uh(θ|ef, sf) lies
entirely below that of u`(θ) because sf ≥ wf, whence no women would pursue
promotion. Also, if ef = 0, the firm’s expected profit from promoting any
woman would be 1
2(1 − sf)(1 + D(sf)) = 1
2(1 − sf)(1 + αsf) < 0.07 since she would leave if her type exceeds sf ≥ wf > 0.9, rendering women much less
productive resource than men who can generate a surplus of 0.5 by Lemma 2. These cases do not arise in equilibrium and have little bearing on our analysis, so will no longer be discussed. Hence, we consider sf > wf below.
Next, if uh(θ|ef, sf) > u`(θ) for all θ ≥ 0 so that the graph of uh(θ|ef, sf)
lies entirely above that of u`(θ) then women of all types pursue promotion.
In this case we define the threshold (type), which is denoted by θc and codes
the marginal type women of types below which pursue promotion, to be ∞. Also, if the upward-sloping part of uh(θ|ef, sf) falls on that of u`(θ), women
of types θ < sf definitely pursue promotion (since sf > wf) whilst women
of all other types are indifferent and some of them may pursue promotion (and leave at midpoint). However, the exact subset of types who pursue promotion only to depart later, is inconsequential so long as the measure of such a subset, say ζ ≥ 0, is unchanged because they do not affect any other aspect of the equilibrium. Thus, we assume without loss of generality that women pursue promotion if and only their types are below the threshold θc
defined by D(θc) = D(s
Finally, the remaining possibility is that the graph of uh(θ|ef, sf) either
crosses that of u`(θ) from above at exactly one point, or overlaps with it for
θ ≤ wf and lies below it for θ > wf.15 In the former case, women pursue
pro-motion if and only if their types are below the crossing point which therefore constitutes the threshold, defined by
θc(e
f, sf) = 2sf − wf − (ef)2. (3)
In the latter case, only women whose types are below wf may pursue
pro-motion out of indifference and exactly who do so is inconsequential for the same reason as above. So, we assume that women pursue promotion if and only their type is below the threshold θc defined by D(θc) being equal to the
fraction of women that pursue promotion in equilibrium.
Note that our notion of threshold type enables us to consider only the “cutoff-style” equilibria of (si, µi)-subgames in which women pursue
promo-tion if and only if their types are below a certain threshold θc ≥ 0. If θc> s f,
some women of types above sf pursue promotion only to leave afterwards.
Such equilibria of (si, µi)-subgames are referred to as equilibria “with
depar-tures.” However, these do not arise along the equilibrium path. In all other equilibria of (si, µi)-subgames, i.e., with θc≤ s
f, any woman who chooses to
pursue promotion will never leave the post, which we refer to as “departure-free” equilibria.
We now examine the possible equilibrium surplus of a firm per female pro-motion. Focusing on departure-free equilibria with threshold θc ∈ (w
f, 1/α),
the firm’s surplus per female promotion is bounded above by max
ef≥0,sf≥θc
1 + ef − sf s.t u`(θc) = uh(θc|ef, sf) (4)
where the constraint is equivalent to ef =
p
2sf − wf − θc since sf ≥ θc> wf.
It is straightforwardly verified that if θc≤ 1 + w
f then sf ≥ θc does not bind
and the solution to (4) is (ef, sf) = (1,1+wf+θ
c
2 ); If θc> 1 + wf, on the other hand, the constraint sf ≥ θcbinds, so the solution is (ef, sf) = (
p
θc− w f, θc).
Therefore, the optimized per-promotion surplus is 1.5 −wf+θc
2 for θc≤ 1 + wf and 1 +pθc− w
f − θc for θc> 1 + wf and thus, it exceed the per-promotion
surplus from men, 0.5, if and only if θc< ¯θ := 2 − w
f < 1 + wf.
If θc ≤ w
f in departure-free equilibria, sf − (ef)2/2 = wf as discussed
above. In this case, by solving maxef,sf 1+ef−sf subject to sf−(ef)
2/2 = w
f,
we get the maximum per-promotion surplus of 1.5 − wf, which is obtainable
when ef = 1 and sf = 0.5 + wf. Since per-promotion surplus is lower in
equilibria with departures (as shown in Appendix), we have the next lemma.
15The two graphs coincide everywhere if s
Lemma 3. Fix an arbitrary θ ∈ (0, ¯θ] where ¯θ = 2 − wf. The maximum
possible surplus per female promotion in equilibria with threshold θ is πf∗(θ) :=
½
1.5 − wf for θ < wf,
1.5 − wf+θ
2 ≥ 0.5 for θ ∈ [wf, ¯θ] with strict inequality if θ < ¯θ.
This maximum is achieved only in departure-free equilibria in which ef = 1
and s∗f(θ) :=
½
0.5 + wf < 1.5 for θ < wf,
1+wf+θ
2 ≤ 1.5 for θ ∈ [wf, ¯θ] with strict inequality if θ < ¯θ.
Note that π∗
f(θ) + s∗f(θ) = 2. In particular, πf∗(θ) strictly decreases in θ ∈
[wf, ¯θ] from πf∗(wf) = 1.5 − wf > 0.5 down to πf∗(¯θ) = 0.5. If the equilibrium
threshold type is θ > ¯θ, the per-promotion surplus is less than 0.5. Proof. See Appendix.
A graphical illustration is useful. Consider θc ∈ (w
f, ¯θ) depicted in the
first diagram of Figure 3. The graph of u`(θc) + c(ef) in the second diagram is
the set of all pairs (ef, sf) that would induce the threshold θc as per (3). The
firm’s surplus, 1+ef−sf, is depicted by the horizontal distance between 1+ef
and u`(θc) + c(ef) and thus, obtains its maximum value πf∗(θc) =
3−wf−θc
2 at
ef = 1, i.e., when efficiency is achieved. At the optimum, s∗f(θc) = 2−πf∗(θc) =
1+wf+θc
2 > θc because 1 + wf > ¯θ > θc, i.e., the flat part of the graph of
uh(θ|ef, sf) crosses the upward-sloping part of the graph of u`(θ) as depicted
in the first diagram, hence no promoted women leave their posts mid-career. [Figure 3 about here]
We are now ready to examine a monopoly firm’s optimal hiring between male and female workers. We begin with the following observation.
Lemma 4: In any continuation equilibrium after a firm offers a contract si = (s, s) where s > 1.1, all promoted workers will have exerted the same
effort e∗ =√2s − 2 > 0 and will never leave their posts.
Proof: Since men would pursue promotion only if s − e2
m/2 ≥ 1, i.e.,
em ≤ e∗ =
√
2s − 2, where em is the effort level they exert for promotion,
1 + e∗− s is the maximum possible surplus per male promotion.
First, consider the case that 1 + e∗ − s > 0, so that s ∈ (1.1, 3). Let
ef denote the effort level exerted by any promoted women, which is unique
if exists by Lemma 1. If ef > e∗ in any equilibrium after si = (s, s) is
offered, then i) no men would be promoted since otherwise women could have guaranteed promotion by exerting e ∈ (em, ef) due to (Ci) given ef >
e∗ ≥ e
m and thus, ii) all upper-tier posts should be filled by women since
otherwise women could have guaranteed promotion by exerting e ∈ (e∗, e f)
due to (Cii). This would require that at least one half of women get promoted, i.e., u`(2α1 ) ≤ uh(2α1 |ef, s) ⇒ ef ≤ max{
q
2s − wf − 2α1 ,√s − wf} < e∗ where
the inequality follows from s > 1.1, given wf ∈ (0.9, 1) and α ∈ (0, 1/4),
contradicting ef > e∗. Thus, we deduce that ef ≤ e∗ in any equilibrium after
si = (s, s) is offered. Consequently, the maximum possible total surplus of
the firm is 1 + e∗− s, which is feasible only if all promoted workers will have
exerted e∗ and measure 0 of them leave their posts.
Hence, it suffices to show that a continuation equilibrium exist in which the firm’s total surplus is 1 + e∗ − s. To do this, note from (H1)-(H2) that
the firm can ensure to hire some measure µi
m ≥ 1 of men and measure µif =
1 − µi
m of women. Then, not all upper-tier posts are filled by women: If
they were, we would have u`(1α) ≤ uh(α1|ef, s) which would imply ef < e∗
since s ∈ (1.1, 3) given wf ∈ (0.9, 1) and α ∈ (0, 1/4), whence men would
guarantee promotion by exerting some e ∈ (ef, e∗) by (Ci). In light of (Cii),
this means that some men get promoted in equilibrium after exerting em ≤
e∗. Thus, women of sufficiently low θ should get promoted as well due to
(Ci) because u`(0) = wf < 1 ≤ uh(0|em, s). Given µim ≥ 1, this means
that men should be indifferent between pursuing promotion and not, i.e.,
em = e∗ and consequently, ef = e∗ since otherwise (Ci) would dictate that
some workers can benefit by exerting slightly less effort. Hence, in the unique continuation equilibrium women exert e∗ if and only if their types are below
θc(e∗, s) = 2 − w
f < 1.1 < s and measure 1 − µifD(θc(e∗, s)) < 1 of men exert
e∗. Consequently, the firm’s total surplus is 1 + e∗− s, as desired.
Next, suppose 1 + e∗− s ≤ 0, so that s ≥ 3. Then, any surplus of the firm
would come from female promotion. If 1 + ef − s > 0 where ef is the effort
level that promoted women will have exerted, then 2[u`(2α1 ) − uh(2α1 |ef, s)] =
min{wf+2α1 −2s, wf−s}+e2f > min{wf+2α1 −2s, wf−s}+(s−1)2 > 0 where
the last inequality ensues because s ≥ 3, wf ∈ (0.9, 1) and α ∈ (0, 1/4), which
would imply that less than one half of women pursue promotion. Since men would not exert more than e∗ ≤ s − 1 < e
f, this would contradict ef being
equilibrium effort level because women would guarantee promotion by exerting
e ∈ (e∗, e
f) due to (Ci) or (Cii). Hence, we conclude that 1 + ef − s ≤ 0 and
thus, any promoted worker will have exerted e = s − 1. No promoted women would leave because 2[u`(s) − uh(s|s − 1, s)] = wf + s2− 3s + 1 > 0.
Lemma 4 highlights a key aspect of our framework: Firms know that offering women the same promotion deal as men would be enough to erase all differences between the two genders (unless they offer very inefficient deals, i.e., s < 1.1, which is easily verified to be suboptimal), in the sense that all promoted women would exert the same effort level as men, and would
remain in their posts with certainty throughout period 2. These women would therefore be identical to their male counterparts in all respects. We stress that this result holds for firms engaged in Bertrand competition as well, since the proof does not rely on the firm being a monopolist.
This observation is crucial for interpreting our results. In particular, it shows that, if any difference between male and female promotion contracts persists in equilibrium, this cannot be justified by invoking differences in the two genders’ propensity to leave their high-rank jobs. Our model therefore allows us to investigate whether the differential treatment suffered by women with respect to promotion may survive independently of an often-invoked argument, namely that firms are less eager to promote women because women are more likely to leave their posts afterwards.
We characterize a monopoly firm’s optimal strategy below, which indeed treats women differently from men. Recall from Lemma 2 that this firm, i, can extract the maximum possible per-promotion surplus of 0.5 from men by offering sm = 1.5 and inducing em = 1. In fact, a firm can guarantee an
equilibrium in which all upper-tier posts generate a surplus of 0.5 by hiring measure 2 of men after offering a male salary sm = 1.5.16 Thus, the firm may
benefit by promoting women only if per-promotion surplus exceeds 0.5 for women. By Lemma 3, this is feasible in equilibria with threshold θ ∈ (0, ¯θ).
In such an equilibrium of (si, µi)-subgame, women pursue promotion if
and only if their types are below the threshold, say θ, and hence, the measure of promoted women is µfαθ < 1 where the inequality follows because µf ≤
2, α ≤ 1/4 and ¯θ < 2. The remaining upper-tier posts (of measure 1 − µfαθ)
may be filled by measure 2 − µf of men to the extent possible. Letting πg
denote the equilibrium per-promotion surplus for gender g workers, therefore, the firm’s total surplus is no higher than πfµfαθ + πmmax{2 − µf, 1 − µfαθ}
which, in light of the upper bounds of πm and πf identified in Lemmas 2 and
3, achieves the maximum value of ΠL(θ) := 0.5 +
D(θ)(π∗
f(θ) − 0.5)
1 − D(θ) (5)
when µf = 1/(1 − D(θ)) ∈ (1, 2), πf = πf∗(θ) and πm = 0.5. Indeed, for
threshold θ ∈ (wf, ¯θ), the firm obtains a total profit equal to this upper bound
in the unique equilibrium of the continuation subgame ensuing the firm’s offer of a contract (sm, sf) = (1.5, s∗f(θ)), as stated in the next lemma.
Lemma 5. (a) A monopoly firm i’s total surplus in any equilibrium with
threshold θ ≤ ¯θ is bounded above by ΠL(θ), which strictly increases in θ ≤ wf 16Exactly measure 1 of men pursue promotion by exerting e = 1 in equilibrium: If less
than measure 1 were to pursue, deviation by exerting an effort level slightly less than 1 would warrant promotion by court protection (Cii) and thus, beneficial.
and is strictly concave in θ ∈ [wf, ¯θ] with a unique maximum at
ˆ
θ = arg max
wf≤θ≤¯θ
ΠL(θ) < ¯θ. (6)
(b) For θ ∈ [wf, ¯θ], (s∗(θ), µ)-subgame has a unique equilibrium if
µf ≤ µ∗f(θ) := 1/(1 − D(θ)) where s∗(θ) := (1.5, s∗f(θ)). (7)
In this equilibrium, measure 1−µfD(θ) of men and all women of types below θ
exert e = 1 and get promoted, so the firm’s surplus, 0.5+µfD(θ)(π∗f(θ)−0.5),
strictly increases in µf from 0.5 when µf = 0 to ΠL(θ) when µf = µ∗f(θ).
(c) For θ ∈ [wf, ¯θ], the firm’s surplus is strictly lower than ΠL(θ) in any
equilibrium of any (s∗(θ), µ)-subgame if µ
f > µ∗f(θ).
Proof. See Appendix.
Lemma 5 establishes that a firm optimally extracts a higher surplus from promoting women compared with men, by offering an inferior contact. This happens because the utility that women of a sufficiently low type θ can expect to obtain if they forgo promotion, u`(θ), is smaller than 1, the utility that a
man can obtain in the same contingency. Intuitively, by forgoing promotion and therefore entering the low-tier job market, low-type women would be pooled with high-type women who would depart for their outside option, and would therefore be paid a low wage that reflects the departure risk of the pool. Since the utility that they can expect to earn if they forgo promotion is lower than that of men, low-type women are therefore willing to pursue promotion even if their deal is inferior to that offered to men. Note that this result depends on women’s type being their private information. The inability of employers to observe a woman’s type is thus a key ingredient of our analysis. Not also that, faced with an unconstrained labor force, a firm would opti-mally hire (and promote) a positive mass of both men and women even if the profit that the firm earns from each promoted woman exceeds what it earns from promoting a man. Conditional on a particular threshold type θ, hiring too many women, i.e., more than measure µ∗
f(θ) of them, would result in too
few workers pursuing promotion: Out of all women hired by the firm, only a fraction D(θ) < D(¯θ) < 0.5 decide to go for promotion, resulting in some
upper-tier posts remain unfilled even after promoting all men. Thus, the firm would benefit by hiring more men so as to be able to fill up all remaining upper-tier posts with men, which happens when µf = µ∗f(θ). On the other
hand, if too few women were to be hired, i.e., if µf < µ∗f(θ), then the firm
could increase its profits by hiring more women so as to fill a larger fraction of upper-tier posts by “higher-yielding” women.
Lemma 5 leads to the equilibrium outcome of a monopoly firm described in Theorem 1 below, for which we need the following technical result:
Lemma 6. Any (si, µi)-subgame has an equilibrium.
Proof. See Appendix.
Theorem 1. If there is a monopoly firm, in the unique equilibrium
out-come the firm maximizes its total surplus by offering sm = 1.5 and sf = s∗f(ˆθ)
and then hiring measure µ∗
f(ˆθ) ∈ (1, 2) of women and measure 2−µ∗f(ˆθ) of men;
Women of types below ˆθ and all men exert e = 1 and get promoted.
Proof. In any equilibrium of any (si, µi)-subgame, the firm’s total surplus
is no higher than 0.5 if the threshold exceeds ¯θ. Thus, Theorem 1 follows from
Lemmas 5 and 6.
It is evident from Theorem 1 that women are treated unfavorably in pro-motion: They have to work as hard as men to be promoted even if they know that they will be paid strictly less after promotion (s∗
f(ˆθ) < sm = 1.5). This
treatment is unjustified since, as shown in Lemma 4, offering sf = sm would
be sufficient to ensure that promoted women exert the same effort as men and remain in their posts with certainty throughout period 2, making them fully identical to men from the employer’s perspective. Part of the rationale for the result is that the monopoly employer capitalizes on its market power to extract a higher surplus from some women who are more desperate than men in pursuing promotion since their alternatives are worse. However, surplus-extraction by a monopoly employer is only part of the story. The next section characterizes the equilibrium offers when firms engage in Bertrand competi-tion for employees. Most surprisingly, we show that the differential treatment suffered by promoted women survives even in this case.
4. Unique Equilibrium under Bertrand Competition
In light of the hiring rules in the presence of a rival firm, (H1) and (H2), each firm’s critical concern is whether to try to assume the leader status by offering a contract of a higher value than the rival’s. Recall that the value
v(s) of a contract s is the ex ante utility of a female worker in the continuation
equilibrium most favored by the firm, following the firm’s offer of s. For s∗(θ)
where θ ∈ [wf, ¯θ], therefore, v(s∗(θ)) = v(θ) := u `(θ)D(θ) + Z θ0>θ u`(θ0)dD(θ0) (8)
because, by Lemma 5, measure µ∗
f(θ) of women are hired and they pursue
promotion when their types are below θ in the unique equilibrium of the continuation subgame ensuing an offer of s∗(θ). Note that, abusing notation
slightly, we also use v(·), as a function of θ, to represent ex ante utility of a female worker in a departure-free equilibrium with threshold θ. Clearly,
v(θ) is constant at v(wf) for θ ≤ wf and strictly increases for θ ≥ wf. That
v(s) ≥ v(wf) for all s is trivial since women get v(wf) by forgoing promotion.
It is intuitive (and proved in Appendix) that offering a contract of a value larger than v(¯θ) is suboptimal because the maximum possible surplus thereof
falls short of 0.5 which can be guaranteed by s∗(θ) for θ ∈ (w
f, ¯θ) according to
Lemma 5. Thus, we focus on contracts with values between v(wf) and v(¯θ).
For θ ∈ [wf, ¯θ], Lemma 5 (b) and (c) establish that the unique equilibrium
ensuing a firm’s offer of s∗(θ), conditional on the firm becoming a leader,
de-livers ΠL(θ) to the firm, the maximum possible surplus subject to offering a
contract of value v(θ). Furthermore, if an equilibrium with threshold θ gen-erates a surplus of ΠL(θ), the equilibrium contract must be s∗(θ) by Lemmas
2 and 3 and the discussion surrounding derivation of (5). Consequently, s∗(θ)
is the uniquely optimal contract for a firm to offer among all contracts whose value is v(θ), conditional on the firm becoming a leader.17 Since s∗(θ) is an
optimal contract among those with the same value, v(θ), conditional on the firm becoming a follower as well (as is shown in Appendix), we have
Lemma 7. In any equilibrium, both firms offer a contract in S∗ = { s∗(θ) | w
f ≤ θ ≤ ¯θ }. (9)
Proof. See Appendix.
It is clear from Lemma 5 that both firms would have liked to obtain the maximum possible surplus, ΠL(ˆθ) = maxθΠL(θ), by hiring measure µf(ˆθ) > 1
of women after offering s∗(ˆθ), but this is not possible because hiring more than
half of the total women workforce requires the firm being a leader in hiring, and not both firms can do so. Furthermore, Bertrand competition implies that achieving ΠL(θ) for any other θ 6= ˆθ by offering s∗(θ) and acquiring the
leader status, would not be viable in equilibrium unless the other firm can also obtain at least the same payoff as a follower, for otherwise the other firm would snatch the leader status by “overbidding” marginally.
In order to see when this may be the case, from Lemma 5 we derive the surplus of a firm as a follower when it has offered s∗(θ0), i.e, conditional on the
other firm having offered s∗(θ) where θ ∈ (θ0, ¯θ] and hired measure µ∗
f(θ) > 1
of women as per Lemma 5, leaving measure 2 − µ∗
f(θ) of them for the follower:
ΠF(θ|θ0) := 0.5 +
1 − 2D(θ) 1 − D(θ) D(θ
0)(π∗
f(θ0) − 0.5). (10) 17However, s∗(θ) is not optimal among all contracts such that s
f = s∗f(θ). For example, if θ > ˆθ then increasing sm may lower the threshold so that ΠL(θ) is higher (hence lower
v), and consequently leads to a higher total surplus, albeit it is not optimal for the reduced
